Elsevier

Discrete Mathematics

Volume 313, Issue 22, 28 November 2013, Pages 2638-2649
Discrete Mathematics

On 1-improper 2-coloring of sparse graphs

https://doi.org/10.1016/j.disc.2013.07.014Get rights and content
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Abstract

A graph G is (1,1)-colorable if its vertices can be partitioned into subsets V1 and V2 such that every vertex in G[Vi] has degree at most 1 for each i{1,2}. We prove that every graph with maximum average degree at most 145 is (1,1)-colorable. In particular, it follows that every planar graph with girth at least 7 is (1,1)-colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to 145 (from above) that are not (1,1)-colorable.

In fact, we establish the best possible sufficient condition for the (1,1)-colorability of a graph G in terms of the minimum, ρG, of ρG(S)=7|S|5|E(G[S])| over all subsets S of V(G). Namely, every graph G with ρG0 is (1,1)-colorable. On the other hand, we construct infinitely many non-(1,1)-colorable graphs G with ρG=1. This solves a related conjecture of Kurek and Ruciński from 1994.

Keywords

Improper coloring
Sparse graph
Maximum average degree
Planar graph

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